On a reaction-diffusion-advection system: fixed boundary or free boundary

This paper is devoted to the asymptotic behaviors of the solution to a reaction-diffusion-advection system in a homogeneous environment with fixed boundary or free boundary. For the fixed boundary problem, the global asymptotic stability of nonconstant semi-trivial states is obtained. It is also shown that there exists a stable nonconstant co-existence state under some appropriate conditions. Numerical simulations are given not only to illustrate the theoretical results, but also to exhibit the advection-induced difference between the left and right boundaries as time proceeds. For the free boundary problem, the spreading-vanishing dichotomy is proved, i.e., the solution either spreads or vanishes finally. Besides, the criteria for spreading and vanishing are further established.